Replace initial Route in MaterialApp without animation?

What you're doing sounds somewhat like a BottomNavigationBar, so you might want to consider one of those instead of a Drawer.

However, having a single Scaffold and updating the body when the user taps a drawer item is a totally reasonable approach. You might consider a FadeTransition to change from one body to another.

Or, if you like using Navigator but don't want the default slide animation, you can customize (or disable) the animation by extending MaterialPageRoute. Here's an example of that:

import 'package:flutter/material.dart';

void main() {
  runApp(new MyApp());
}

class MyCustomRoute<T> extends MaterialPageRoute<T> {
  MyCustomRoute({ WidgetBuilder builder, RouteSettings settings })
      : super(builder: builder, settings: settings);

  @override
  Widget buildTransitions(BuildContext context,
      Animation<double> animation,
      Animation<double> secondaryAnimation,
      Widget child) {
    if (settings.isInitialRoute)
      return child;
    // Fades between routes. (If you don't want any animation, 
    // just return child.)
    return new FadeTransition(opacity: animation, child: child);
  }
}

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return new MaterialApp(
      title: 'Navigation example',
      onGenerateRoute: (RouteSettings settings) {
        switch (settings.name) {
          case '/': return new MyCustomRoute(
            builder: (_) => new MyHomePage(),
            settings: settings,
          );
          case '/somewhere': return new MyCustomRoute(
            builder: (_) => new Somewhere(),
            settings: settings,
          );
        }
        assert(false);
      }
    );
  }
}

class MyHomePage extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      appBar: new AppBar(
        title: new Text('Navigation example'),
      ),
      drawer: new Drawer(
        child: new ListView(
          children: <Widget> [
            new DrawerHeader(
              child: new Container(
                  child: const Text('This is a header'),
              ),
            ),
            new ListTile(
              leading: const Icon(Icons.navigate_next),
              title: const Text('Navigate somewhere'),
              onTap: () {
                Navigator.pushNamed(context, '/somewhere');
              },
            ),
          ],
        ),
      ),
      body: new Center(
        child: new Text(
          'This is a home page.',
        ),
      ),
    );
  }
}

class Somewhere extends StatelessWidget {
  Widget build(BuildContext context) {
    return new Scaffold(
      body: new Center(
        child: new Text(
          'Congrats, you did it.',
        ),
      ),
      appBar: new AppBar(
        title: new Text('Somewhere'),
      ),
      drawer: new Drawer(
        child: new ListView(
          children: <Widget>[
            new DrawerHeader(
              child: new Container(
                child: const Text('This is a header'),
              ),
            ),
          ],
        ),
      ),
    );
  }
}

Use PageRouteBuilder like:

Navigator.push(
  context,
  PageRouteBuilder(
    pageBuilder: (_, __, ___) => Screen2(),
    transitionDuration: Duration.zero,
  ),
);

---

And if you want transition, simply add following property to above PageRouteBuilder, and change seconds to say 1.

transitionsBuilder: (_, a, __, c) => FadeTransition(opacity: a, child: c),